On generalized metric spaces: a survey (Q2878098)

One of the interesting generalization of the notion of metric space was introduced by \textit{A. Branciari} in [Publ. Math. 57, No.~ 1--2, 31--37 (2000; Zbl 0963.54031)] where the triangle inequality was replaced by a so called quadrilateral inequality. It was not immediately observed that such spaces, called generalized metric spaces, may fail to satisfy some standard metric properties. Hence in some of the first papers, the authors implicitly used some of these conditions, so that their proofs were flawed. \textit{B. Samet} [Publ. Math. 76, No. 4, 493--494 (2010; Zbl 1224.54106)] and \textit{I. R. Sarma} et al. [J. Nonlinear Sci. Appl. 2, No. 3, 180--182 (2009; Zbl 1173.54311)] presented examples that there exist generalized metric spaces that do not satisfy standard metric properties. As a consequence, most of the authors dealing with such spaces made some additional requirements in order to deduce their results. In this paper, the authors prove that such requirements are usually superfluous since some easy results can be used to compensate the lack of the above mentioned standard properties of metric spaces.